Seismic moment distribution revisited: I. Statistical results

Abstract

SUMMARY An accumulation of seismic moment data gathered over the previous decade justifies a new attempt at a comprehensive statistical analysis of these data: herein, more rigourous statistical techniques are introduced, their properties investigated, and these methods are employed for analysis of large modern data sets. Several theoretical distributions of earthquake size (seismic moment–frequency relations) are described and compared. We discuss the requirements for such distributions and introduce an upper bound or a ‘corner moment’ for a distribution to have a finite energy or moment flux. We derive expressions for probability density functions and statistical moments of the distributions. We also describe parameter evaluation, in particular how to estimate the seismic moment distribution for the largest earthquakes. Simulating earthquake size distributions allows for a more rigourous evaluation of distribution parameters and points to the limitations of the classical statistical analysis of earthquake data. Simulations suggest that several earthquakes approaching or exceeding the corner magnitude (mc) limit need to be registered to evaluate mc with reasonable accuracy. Using the Harvard catalogue data, we compare moment distribution parameters for various temporal spans of the catalogue, for different tectonic provinces and depth ranges, and for earthquakes with various focal mechanisms. The statistical analysis suggests that the exponent β is universal (β=0.60–0.65) for all moderate earthquakes. The corner moment (Mc) value, determined by the maximum‐likelihood method, both in subduction zones and globally, is about 1021 N m, corresponding to the corner moment magnitude mc≈8.0. For mid‐oceanic earthquakes, mc is apparently smaller for spreading ridges, it is about 5.8, and for strike‐slip earthquakes on transform faults it decreases from 7.2 to 6.5 as the relative slip velocity of faults increases. We investigate the seismic moment errors, both random and systematic, and their dependence on earthquake size. The relative errors seem to decrease for larger events. The influence of moment uncertainties on the parameter estimates is studied. Whereas the β values do not appear to be significantly influenced by the errors, for the corner moment large errors can lead to substantially biased estimates. We compare the Harvard catalogue results with the earthquake data from instrumental catalogues in the first three‐quarters of the 20th century. Several very large earthquakes (m≥9) occurred around the middle of the century. Their magnitude values cannot be fitted by a modified Gutenberg–Richter law with mc=8.0–8.5. Among other factors, this discrepancy can be explained by either substantially higher errors in the earlier magnitude values, or by mc being higher for some subduction zones. It is unlikely that data available presently or soon will be sufficient to determine the corner magnitude of 9 and above, with reasonable precision, using purely statistical methods.

Probabilistic Analysis of Tsunami Hazards*

“…For example, the maximum earthquake magnitude that any given fault can generate may be subject to different interpretations of, for example, fault length, seismogenic depth, and slip rate. More generally, different frequency-magnitude curves can be envisioned to conform to either a modified G-R relationship in which the b-value is greater for larger earthquakes (Pacheco et al, 1992;Sornette and Virieux, 1992;Romanowicz and Rundle, 1993;Kagan, 1999;Pisarenko and Sornette, 2004) or a characteristic model for the largest earthquake (Wesnousky, 1994;Kagan, 2002b). If hazard analysis is based on the seismic moment (M 0 =l LWD) of potential earthquakes, then values and associated uncertainties for first-order parameters such as rupture length (L), width (W), shear modulus (l), and average slip (D) must be provided.…”

Assessment of tsunami hazards for the Central American Pacific coast from southern Mexico to northern Peru

“…Incidentally, we observe that Eq. (2) exhibits the exponentially tapered functional form recommended by Kagan (2002) (see also the discussion in Geist et al, 2009). The values of a and b obtained for zones 1, 2 and 4 and the magnitude range used for the estimation are shown in Table 2.…”

Integer and fractional-order entropy analysis of earthquake data series

“…where N (m) is the frequency of earthquakes with magnitude greater than m, parameter r describes the relation between small and large earthquake numbers and q is a measure of the seismic activity, or earthquake productivity, of a region [24]. The GR law can also be stated in terms of energy released by earthquakes.…”